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基于离子阱中离子晶体的热传导的研究进展

李冀 陈亮 冯芒

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基于离子阱中离子晶体的热传导的研究进展

李冀, 陈亮, 冯芒

Research progress of heat transport in trapped-ion crystals

Li Ji, Chen Liang, Feng Mang
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  • 热传导现象是物理学中最重要的研究课题之一, 特别是近年来, 随着对单分子器件研究的不断深入, 人们越来越关注低维(一维和二维)微观系统的热传导问题. 离子阱中的离子晶体处于真空环境中, 没有与外部环境进行能量交换, 其晶体结构和温度可以通过电场和光场精确操控, 为研究低维晶体在经典或量子状态下的热传导提供了理想的实验平台. 本文综述了近年来离子晶体中热传导的理论研究, 包括一维、二维和三维模型中温度分布和稳态热流的计算方法, 以及在不同维度离子晶体构型下热流与温度分布的特性. 此外, 还讨论了无序度对离子晶体热导性的影响.
    Heat transport is one of the most important research topics in physics. Especially in recent years, with the in depth study on single-molecule devices, heat transport in low-dimensional (i.e. one- and two-dimensional) microsystems has received more and more attention. In the research of Fermi-Pasta-Ulam crystals and harmonic crystals, it is widely accepted that heat conduction in low-dimensional system does not follow Fourier’s law. Due to the lack of the equipment that can directly measure heat current, it has been proven to be a challenging task to carry out relevant experiments. Ion crystal in ion trap is located in vacuum and does not exchange energy with the external environment. The crystal structure and temperature can be accurately controlled by electric field and optical field, providing an ideal experimental platform for studying thermal conduction in low-dimensional crystals in classical state or quantum state. Herein we summarize the recent theoretical research on thermal conduction in ion crystals, including the methods of calculating temperature distribution and steady-state heat current in one-dimensional, two-dimensional, and three-dimensional models, as well as the characteristics of heat current and temperature distribution under different ion crystal configurations. Because the nonlinear effect caused by the imbalance among three dimensions hinders the heat transport, the heat current in ion crystal is largest in the linear configuration while smallest in the zig-zag configuration. In addition, we also introduce the influence of disorder on the thermal conductivity of ion crystal, including the influence on the heat current across various ion crystal configurations such as the linear, the zig-zag and the helical configuration. Notably, the susceptibility of ion crystal to disorder increases with crystal size increasing. Specifically, the zig-zag ion crystal configuration exhibits the largest susceptibility to disorder, whereas the linear configuration is least affected. Finally, we provide a concise overview of experimental studies of the heat conduction in low-dimensional systems. Examination of the heat conduction in ion crystal offers a valuable insight into various cooling techniques employed in ion trap systems, including sympathetic cooling, electromagnetically induced transparency cooling, and polarization gradient cooling. Just like macroscopic thermal diodes made by thermal metamaterials, it is possible that the microscopic thermal diodes can also be made in low-dimensional systems.
      通信作者: 陈亮, liangchen@wipm.ac.cn ; 冯芒, mangfeng@wipm.ac.cn
    • 基金项目: 国家自然科学基金(批准号: U21A20434)、广州市重点实验室(批准号: 202201000010)和广州市科技计划(批准号: 202201011727)资助的课题.
      Corresponding author: Chen Liang, liangchen@wipm.ac.cn ; Feng Mang, mangfeng@wipm.ac.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. U21A20434), the Key Laboratory of Guangzhou, China (Grant No. 202201000010), and the Science and Technology Project of Guangzhou, China (Grant No. 202201011727).
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    Dhar A 2008 Adv. Phys. 57 457Google Scholar

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    Rieder Z, Lebowitz J L, Lieb E 1967 J. Math. Phys. 8 1073Google Scholar

    [5]

    Dhar A, Dandekar R 2015 Physica A 418 49Google Scholar

    [6]

    Chaudhuri A, Kundu A, Roy D, Dhar A, Lebowitz J L, Spohn H 2010 Phys. Rev. B 81 064301

    [7]

    Saito K, Dhar A 2010 Phys. Rev. Lett. 104 040601Google Scholar

    [8]

    Dhar A 2001 Phys. Rev. Lett. 86 5882Google Scholar

    [9]

    Casher A, Lebowitz J L 1971 J. Math. Phys. 12 1701Google Scholar

    [10]

    Rich M, Visscher W M 1975 Phys. Rev. B 11 2164Google Scholar

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    Rubin R J, Greer W L 1971 J. Math. Phys. 12 1686Google Scholar

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    Verheggen T 1979 Commun. Math. Phys. 68 69Google Scholar

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    Hu B, Li B, Zhao H 2000 Phys. Rev. E 61 3828Google Scholar

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    Wang P, Luan C Y, Qiao M, Um M, Zhang J, Wang Y, Yuan X, Gu M, Zhang J, Kim K 2021 Nat. Commun. 12 233Google Scholar

    [15]

    Kielpinski D, Monroe C, Wineland D J 2002 Nature 417 709Google Scholar

    [16]

    Leibfried D, Blatt R, Monroe C, Wineland D 2003 Rev. Mod. Phys. 75 281Google Scholar

    [17]

    Schiffer J P 1993 Phys. Rev. Lett. 70 818Google Scholar

    [18]

    Eckmann J P, Pillet C A, Rey-Bellet L 1999 Commun. Math. Phys. 201 657Google Scholar

    [19]

    Ford G W, Kac M, Mazur P 1965 J. Math. Phys. 6 504Google Scholar

    [20]

    Zürcher U, Talkner P 1990 Phys. Rev. A 42 3278Google Scholar

    [21]

    Saito K, Takesue S, Miyashita S 2000 Phys. Rev. E 61 2397Google Scholar

    [22]

    Dhar A, Shastry B S 2003 Phys. Rev. B 67 195405Google Scholar

    [23]

    Segal D, Nitzan A, Hänggi P 2003 J. Chem. Phys. 119 6840Google Scholar

    [24]

    Lepri S, Livi R, Politi A 2003 Phys. Rep. 377 1Google Scholar

    [25]

    Lin G D, Duan L M 2011 New J. Phys. 13 075015Google Scholar

    [26]

    Ruiz A, Alonso D, Plenio M B, del Campo A 2014 Phys. Rev. B 89 214305Google Scholar

    [27]

    Freitas N, Martinez E A, Paz J P 2016 Phys. Scr. 91 013007Google Scholar

    [28]

    Ruiz-García A, Fernández J J, Alonso D 2019 Phys. Rev. E 99 062105

    [29]

    Gardiner C, Zoller P 2010 Quantum Noise: A Handbook and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (3rd Ed.) (Berlin: Springer) pp42−89

    [30]

    Walls D F, Milburn G J 2010 Quantum Optics (2nd Ed.) (Berlin: Springer) pp112−117

    [31]

    Novikov E A 1965 Sov. Phys. JETP 20 1290

    [32]

    Plenio M B, Huelga S F 2008 New J. Phys. 10 113019Google Scholar

    [33]

    Kubo R 1966 Rep. Prog. Phys. 29 255Google Scholar

    [34]

    Piccirelli R A 1968 Phys. Rev. 175 77Google Scholar

    [35]

    Martinez E A, Paz J P 2013 Phys. Rev. Lett. 110 130406Google Scholar

    [36]

    Freitas N, Paz J P 2014 Phys. Rev. E 90 042128Google Scholar

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    Freitas N, Paz J P 2014 Phys. Rev. E 90 069903Google Scholar

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    Birkl G, Kassner S, Walther H 1992 Nature 357 310Google Scholar

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    Dubin D H E, O’Neil T M 1999 Rev. Mod. Phys. 71 87Google Scholar

    [42]

    Pyka K, Keller J, Partner H L, Nigmatullin R, Burgermeister T, Meier D M, Kuhlmann K, Retzker A, Plenio M B, Zurek W H, del Campo A, Mehlstäubler T E 2013 Nat. Commun. 4 2291Google Scholar

    [43]

    Yan L L, Wan W, Chen L, Zhou F, Gong S J, Tong X, Feng M 2016 Sci. Rep. 6 21547Google Scholar

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    Li J, Yan L L, Chen L, Liu Z C, Zhou F, Zhang J Q, Yang W L, Feng M 2019 Phys. Rev. A 99 063402Google Scholar

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    Liu Z C, Chen L, Li J, Zhang H, Li C B, Zhou F, Su S L, Yan L L, Feng M 2020 Phys. Rev. A 102 033116Google Scholar

    [46]

    Srivathsan B, Fischer M, Alber L, Weber M, Sondermann M, Leuchs G 2019 New J. Phys. 21 113014Google Scholar

    [47]

    Ramm M, Pruttivarasin T, Häffner H 2014 New J. Phys. 16 063062Google Scholar

    [48]

    Mao Z C, Xu Y Z, Mei Q X, Zhao W D, Jiang Y, Cheng Z J, Chang X Y, He L, Yao L, Zhou Z C, Wu Y K, Duan L M 2022 Phys. Rev. A 105 033107Google Scholar

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    Zuo Y N, Han J Z, Zhang J W, Wang L J 2019 Appl. Phys. Lett. 115 061103Google Scholar

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    Li M, Zhang Y, Zhang Q Y, Bai W L, He S G, Peng W C, Tong X 2023 Chin. Phys. B 32 036402Google Scholar

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    Qiao M, Wang Y, Cai Z, Du B, Wang P, Luan C, Chen W, Noh H R, Kim K 2021 Phys. Rev. Lett. 126 023604Google Scholar

    [52]

    Joshi M K, Fabre A, Maier C, Brydges T, Kiesenhofer D, Hainzer H, Blatt R, Roos C F 2020 New J. Phys. 22 103013Google Scholar

    [53]

    Tighe T S, Worlock J M, Roukes M L 1997 Appl. Phys. Lett. 70 2687Google Scholar

    [54]

    Kim P, Shi L, Majumdar A, McEuen P L 2001 Phys. Rev. Lett. 87 215502Google Scholar

    [55]

    Chang C W, Okawa D, Majumdar A, Zettl A 2006 Science 314 1121Google Scholar

    [56]

    Simón M A, Martínez-Garaot S, Pons M, Muga J G 2019 Phys. Rev. E 100 032109

  • 图 1  $ \rho $值不同时, 由100个离子组成的离子晶体在稳态下的温度分布[25], 其中离子的温度$ T_{i}^{{\mathrm{s}}} $由平均声子数来表示. 第1个离子和第100个离子与热库相接触, 由图中的红线表示, 其他的离子由蓝线表示

    Fig. 1.  Temperature distribution of an ion crystal composed of 100 ions under different $ \rho $ in stable state, where temperature distribution $ T_{i}^{{\mathrm{s}}} $ measured by the mean thermal phonon number. The 1st and the 100th ions are in contact with the thermal bath, indicated by the red line in the figure, while other ions are indicated by the blue line. The figure is taken from Ref. [25].

    图 2  由$ 101 $个离子组成的离子晶体在稳态下的温度分布[25], 其中离子的温度$ T_{i}^{{\mathrm{s}}} $由平均声子数来表示. 第1个离子和第$ 51 $个离子与热库相接触. 与热库相接触的离子由图中的红线表示, 其他的离子由蓝线表示

    Fig. 2.  Temperature distribution of an ion crystal composed of 101 ions under different $ \rho $ in stable state, where temperature distribution $ T_{i}^{{\mathrm{s}}} $ measured by the mean thermal phonon number. The first ion and the $ 51{\mathrm{st }}$ ion are in contact with the thermal bath. Ions in contact with the thermal bath are indicated by the red line in the figure, while other ions are indicated by the blue line. The figure is taken from Ref. [25].

    图 3  处于不同构型下的离子晶体在x方向上的温度分布 [26], 其中$ \alpha $表示结构相变的序参量. 两个颜色不同的区域中的离子和不同参数的激光相互作用. 插图表示离子晶体的构型

    Fig. 3.  Temperature distribution of ion crystals vs. the distance x in different configurations, where $ \alpha $ represents the order parameter of structural phase transition, and the two regions with different colors represent ions interacting with the laser. The insets show the configurations of ion crystals. The figure is taken from Ref. [26].

    图 4  处于不同构型下的离子晶体的稳态热流 [26], 其中$ \alpha $表示结构相变的序参量. 插图表示离子晶体的构型

    Fig. 4.  Heat flux of ion crystals in different configurations, where $ \alpha $ represents the order parameter of structural phase transition. The insets show the configurations of ion crystals. The figure is taken from Ref. [26]

    图 5  在不同$ n_{y} $和$ n_{z} $的取值下, 离子晶体结构相变的相图[28]. 用L (linear), Z (zigzag)和H (helical)这三个字母代表离子晶体的直线构型、之字构型和螺旋构型. 沿HL线, $ (n_{y}, n_{z}) $值的变化会产生直线构型和螺旋构型之间的结构相变. 沿ZL线, $ (n_{y}, n_{z}) $值的变化会产生直线构型与之字构型之间的结构相变. 沿ZHZ线, $ (n_{y}, n_{z}) $值的变化会产生从之字构型到螺旋构型再到之字构型的结构相变. 图中的灰色区域代表直线构型. 在灰色区域之外, HL线之上的区域代表处于$ x{\text{-}}y $平面上的之字构型, 在HL线之下的区域代表处于x -z平面上的之字构型, HL线上为螺旋构型

    Fig. 5.  Phase diagram of ion crystal structural phase transitions under different values of $ n_{y} $ and $ n_{z} $. L (linear), Z (zigzag), and H (helical) represent the linear, zigzag, and helical configurations of ion crystal. ZL line, HL line and ZHZ line represent the zigzag-linear transition, helical-linear transition and zigzag-helical-zigzag transition. The gray region in the figure represents the linear configuration. Outside of the gray region, the region above the HL line represents the zigzag configuration lying in the $ x{\text{-}}y $ plane, while the region below the HL line represents the zigzag configuration lying in the $ x{\text{-}}z $ plane. The region on the HL line represents the helical configuration. The figure is taken from Ref. [28]

    图 6  在离子数$ N=30 $的离子晶体中, 不同构型的离子晶体的稳态温度分布[28], 图中L代表直线构型、Z代表之字构型、H代表螺旋构型, 图中的参数为$ (n_{y}, n_{z}) $ (a) 3种不同构型的离子晶体在稳态下的温度分布, 两个颜色不同的区域中的离子和不同参数的激光相互作用; (b)在稳态下, 3种不同构型的离子晶体不与激光接触的部分的温度梯度分布, 其中$\varDelta_{\rm T} $代表与系统中心的距离

    Fig. 6.  Temperature distribution of ion crystals with 30 ions with different configurations in stable state, where L represents linear configuration, Z represents zigzag configuration, and H represents helical configuration. The labels indicate the values of $ (n_{y}, n_{z}) $: (a) Temperature distribution of three different configurations of ion crystals in stable state, where two different colors represent ions interacting with the laser; (b) distribution of temperature gradients in the parts of ion crystals with three different configurations that are not in contact with the laser in stable state, where $ \varDelta_{\rm T} $represents the distance from the center of the system. The figure is taken from Ref. [28].

    图 7  在离子数$ N=30 $的离子晶体中, 离子晶体从处于x-z平面的之字构型相变到螺旋构型再相变到处于x-y平面的之字构型的过程中的温度分布和热流变化[28] (a)离子晶体在结构相变过程中的3种典型的温度分布, 两个颜色不同的区域中的离子和不同参数的激光相互作用, 图中Z代表之字构型, H代表螺旋构型, 图中的参数为$ (n_{y}, n_{z}) $; (b)在$ n_{y} $保存不变时, $ n_{z} $不断增大的过程中的热流变化

    Fig. 7.  Temperature distribution and heat flux changes in an ion crystal with 30 ions during the process of zigzag-helical-zigzag transition where Z represents zigzag configuration and H represents helical configuration: (a) Three typical temperature distributions of the ion crystal during the structural phase transition process, where two different colors represent ions interacting with the laser; the labels indicate the values of $ (n_{y}, n_{z}) $; (b) changes in heat flux during the process of increasing $ n_{z} $ while keeping $ n_{y} $ constant. The figure is taken from Ref. [28].

  • [1]

    Li B, Wang L, Hu B 2002 Phys. Rev. Lett. 88 223901Google Scholar

    [2]

    Dhar A, Roy D 2006 J. Stat. Phys. 125 801Google Scholar

    [3]

    Dhar A 2008 Adv. Phys. 57 457Google Scholar

    [4]

    Rieder Z, Lebowitz J L, Lieb E 1967 J. Math. Phys. 8 1073Google Scholar

    [5]

    Dhar A, Dandekar R 2015 Physica A 418 49Google Scholar

    [6]

    Chaudhuri A, Kundu A, Roy D, Dhar A, Lebowitz J L, Spohn H 2010 Phys. Rev. B 81 064301

    [7]

    Saito K, Dhar A 2010 Phys. Rev. Lett. 104 040601Google Scholar

    [8]

    Dhar A 2001 Phys. Rev. Lett. 86 5882Google Scholar

    [9]

    Casher A, Lebowitz J L 1971 J. Math. Phys. 12 1701Google Scholar

    [10]

    Rich M, Visscher W M 1975 Phys. Rev. B 11 2164Google Scholar

    [11]

    Rubin R J, Greer W L 1971 J. Math. Phys. 12 1686Google Scholar

    [12]

    Verheggen T 1979 Commun. Math. Phys. 68 69Google Scholar

    [13]

    Hu B, Li B, Zhao H 2000 Phys. Rev. E 61 3828Google Scholar

    [14]

    Wang P, Luan C Y, Qiao M, Um M, Zhang J, Wang Y, Yuan X, Gu M, Zhang J, Kim K 2021 Nat. Commun. 12 233Google Scholar

    [15]

    Kielpinski D, Monroe C, Wineland D J 2002 Nature 417 709Google Scholar

    [16]

    Leibfried D, Blatt R, Monroe C, Wineland D 2003 Rev. Mod. Phys. 75 281Google Scholar

    [17]

    Schiffer J P 1993 Phys. Rev. Lett. 70 818Google Scholar

    [18]

    Eckmann J P, Pillet C A, Rey-Bellet L 1999 Commun. Math. Phys. 201 657Google Scholar

    [19]

    Ford G W, Kac M, Mazur P 1965 J. Math. Phys. 6 504Google Scholar

    [20]

    Zürcher U, Talkner P 1990 Phys. Rev. A 42 3278Google Scholar

    [21]

    Saito K, Takesue S, Miyashita S 2000 Phys. Rev. E 61 2397Google Scholar

    [22]

    Dhar A, Shastry B S 2003 Phys. Rev. B 67 195405Google Scholar

    [23]

    Segal D, Nitzan A, Hänggi P 2003 J. Chem. Phys. 119 6840Google Scholar

    [24]

    Lepri S, Livi R, Politi A 2003 Phys. Rep. 377 1Google Scholar

    [25]

    Lin G D, Duan L M 2011 New J. Phys. 13 075015Google Scholar

    [26]

    Ruiz A, Alonso D, Plenio M B, del Campo A 2014 Phys. Rev. B 89 214305Google Scholar

    [27]

    Freitas N, Martinez E A, Paz J P 2016 Phys. Scr. 91 013007Google Scholar

    [28]

    Ruiz-García A, Fernández J J, Alonso D 2019 Phys. Rev. E 99 062105

    [29]

    Gardiner C, Zoller P 2010 Quantum Noise: A Handbook and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (3rd Ed.) (Berlin: Springer) pp42−89

    [30]

    Walls D F, Milburn G J 2010 Quantum Optics (2nd Ed.) (Berlin: Springer) pp112−117

    [31]

    Novikov E A 1965 Sov. Phys. JETP 20 1290

    [32]

    Plenio M B, Huelga S F 2008 New J. Phys. 10 113019Google Scholar

    [33]

    Kubo R 1966 Rep. Prog. Phys. 29 255Google Scholar

    [34]

    Piccirelli R A 1968 Phys. Rev. 175 77Google Scholar

    [35]

    Martinez E A, Paz J P 2013 Phys. Rev. Lett. 110 130406Google Scholar

    [36]

    Freitas N, Paz J P 2014 Phys. Rev. E 90 042128Google Scholar

    [37]

    Freitas N, Paz J P 2014 Phys. Rev. E 90 069903Google Scholar

    [38]

    Tisseur F, Meerbergen K 2001 SIAM Rev. 43 235Google Scholar

    [39]

    Li B, Zhao H, Hu B 2001 Phys. Rev. Lett. 86 63Google Scholar

    [40]

    Birkl G, Kassner S, Walther H 1992 Nature 357 310Google Scholar

    [41]

    Dubin D H E, O’Neil T M 1999 Rev. Mod. Phys. 71 87Google Scholar

    [42]

    Pyka K, Keller J, Partner H L, Nigmatullin R, Burgermeister T, Meier D M, Kuhlmann K, Retzker A, Plenio M B, Zurek W H, del Campo A, Mehlstäubler T E 2013 Nat. Commun. 4 2291Google Scholar

    [43]

    Yan L L, Wan W, Chen L, Zhou F, Gong S J, Tong X, Feng M 2016 Sci. Rep. 6 21547Google Scholar

    [44]

    Li J, Yan L L, Chen L, Liu Z C, Zhou F, Zhang J Q, Yang W L, Feng M 2019 Phys. Rev. A 99 063402Google Scholar

    [45]

    Liu Z C, Chen L, Li J, Zhang H, Li C B, Zhou F, Su S L, Yan L L, Feng M 2020 Phys. Rev. A 102 033116Google Scholar

    [46]

    Srivathsan B, Fischer M, Alber L, Weber M, Sondermann M, Leuchs G 2019 New J. Phys. 21 113014Google Scholar

    [47]

    Ramm M, Pruttivarasin T, Häffner H 2014 New J. Phys. 16 063062Google Scholar

    [48]

    Mao Z C, Xu Y Z, Mei Q X, Zhao W D, Jiang Y, Cheng Z J, Chang X Y, He L, Yao L, Zhou Z C, Wu Y K, Duan L M 2022 Phys. Rev. A 105 033107Google Scholar

    [49]

    Zuo Y N, Han J Z, Zhang J W, Wang L J 2019 Appl. Phys. Lett. 115 061103Google Scholar

    [50]

    Li M, Zhang Y, Zhang Q Y, Bai W L, He S G, Peng W C, Tong X 2023 Chin. Phys. B 32 036402Google Scholar

    [51]

    Qiao M, Wang Y, Cai Z, Du B, Wang P, Luan C, Chen W, Noh H R, Kim K 2021 Phys. Rev. Lett. 126 023604Google Scholar

    [52]

    Joshi M K, Fabre A, Maier C, Brydges T, Kiesenhofer D, Hainzer H, Blatt R, Roos C F 2020 New J. Phys. 22 103013Google Scholar

    [53]

    Tighe T S, Worlock J M, Roukes M L 1997 Appl. Phys. Lett. 70 2687Google Scholar

    [54]

    Kim P, Shi L, Majumdar A, McEuen P L 2001 Phys. Rev. Lett. 87 215502Google Scholar

    [55]

    Chang C W, Okawa D, Majumdar A, Zettl A 2006 Science 314 1121Google Scholar

    [56]

    Simón M A, Martínez-Garaot S, Pons M, Muga J G 2019 Phys. Rev. E 100 032109

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出版历程
  • 收稿日期:  2023-10-28
  • 修回日期:  2023-11-23
  • 上网日期:  2023-12-23
  • 刊出日期:  2024-02-05

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